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The Stacks project

Lemma 88.30.3. Let $A \to B$ be a local map of local Noetherian rings such that

  1. $A \to B$ is flat,

  2. $\mathfrak m_ B = \mathfrak m_ A B$, and

  3. $\kappa (\mathfrak m_ A) = \kappa (\mathfrak m_ B)$

Then the base change functor from the category (88.30.0.1) for $(A, \mathfrak m_ A)$ to the category (88.30.0.1) for $(B, \mathfrak m_ B)$ is an equivalence.

Proof. The conditions signify that $A \to B$ induces an isomorphism on completions, see More on Algebra, Lemma 15.43.9. Hence this lemma is a special case of Lemma 88.30.1. $\square$

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